Questões - IME | Gabarito e resoluções

(IME - 2017/2018 - 2 FASE ) A atmosfera densa de um planeta hipottico possui um ndice de refrao dependente das condies meteorolgicas do local, tais como presso, temperatura e umidade. Considere um modelo no qual a regio da atmosfera formada por K + 1camadas de ndice de refrao diferentes, n0,n1, ..., nkde 1 KMde altura cada, onde o ndice de refrao decai 10%a cada quilmetro de aumento na altitude. Considerando somente os efeitos da reflexo e da refrao na atmosfera, se um raio luminoso, proveniente de um laser muito potente for disparado da superfcie do planeta, formando um ngulo de 60com a tangente superfcie, verifique se este raio alcanar o espao, e, em caso negativo, determine qual ser a altitude mxima alcanada pelo raio. Dados: - o planeta esfrico com raio Rp = 6.370 km; - log10(9) = 0,95e log10(2) = 0,3e - K = 9

(IME - 2017/2018 - 1 FASE ) Conforme a figura acima, duas lanternas muito potentes, cilndricas, com dimetro D = 4 cm, esto alinhadas no plano vertical. Ambas possuem lentes nas extremidades, cujos centros pticos Oesto alinhados verticalmente e cujas distncias focais so f = 3 cm.Uma das lentes convergente e a outra divergente. Suas lmpadas geram raios de luz horizontais, que encontram as lentes das respectivas lanternas e so projetados at um anteparo vertical. Sabendo que a distncia entre os centros pticos das duas lentes y = 12 cm,a distncia mxima x entre os centros pticos das lentes O e o anteparo, em centmetros, que faz com que a luz projetada pelas lanternas no se sobreponha :

(IME - 2017/2018 - 2 FASE )Um circuito de galvanoplastia em escala laboratorial provido de uma fonte eltrica de 110Walimentada por uma rede domstica de 100VO circuito constitudo por trs cubas eletrolticas ligadas em srie que contm, respectivamente, uma soluo de nitrato de prata, uma soluo de sulfato de cobre e 1Lde uma soluo de NaCl 1molar. Pretende-se fazer o recobrimento de uma pea de 100 cm2de superfcie por uma pelcula de prata com 40mde espessura e o recobrimento de outra pea com cobre, alm de eletrolisar uma parte do NaCl Pede-se calcular: a) a massa de cobre que ser depositada na segunda pea; b) a frao de ons Cl-que sero oxidados; c) o tempo requerido para a operao. Dados: Massa especfica da prata: 10,5 g/cm3 Equivalente eletroqumico do cloro: 0,389 mg/C Equivalente eletroqumico da prata: 1,119 mg/C

(IME - 2017/2018 - 2 FASE) A DAY IN THE LIFE OF A NUCLEAR MATERIALS ENGINEER My career _____1_____ a planned one in any way. At school I was athletic; I ran and played badminton to a high standard when I was young and always thought my career would be a sporting one _____2_____ I suffered an injury during my teens. The rest of my family was academic; my father was an aerodynamic engineer and my mother a mathematician, _____3_____ my sister studied geology. At the age of 16, I attended a Women in Science and Engineering careers week with school, just to have a look at what was available. This helped me decide that _____4_____ I really wanted to do was an engineering degree, so I chose to do a BEng in materials science and engineering at Liverpool University, and then went on to do a PhD. My PhD looked at auxetic polymeric materials. No one _____5_____ of them: they get fatter as you stretch them, _____6_____ is very novel, and at the time there were only a handful of researchers in the world working on these. The PhD started my interest in polymeric materials. Towards the end of my PhD I _____7_____ two research roles, and ended up taking a job with British Nuclear Fuels Limited at the Company Research Laboratory (CRL). () During my time at CRL I _____8_____ on secondment to the Sellafield site in Cumbria, which then turned into a permanent position in the research and technology materials and inspection group. During this time I became a chartered engineer and a full professional member of the Institute of Materials Minerals and Mining. I now head up one of Sellafields Centres of Expertise (CoE): I am the CoE lead and subject matter expert for polymeric materials. Recently I _____9_____ as a fellow of the Institute of Materials. I definitely dont have a typical day. I sometimes have a plan, but _____10_____ stick to it as much of my work is responsive to situations which are transient. The range of things I can get involved in is huge and includes specifying materials for use in challenging environments, new plant designs and decommissioning activates. RATHBONE, Penny. Adapted from: The Guardian. A day in the life of a nuclear materials engineer. Disponvel em: https://www.theguardian.com/women-in-leadership/2016/jan/22/a-day-in-the-life-of-a-nuclear-materials-engineer. Acesso em: 22/06/2017. Escolha a alternativa que completa corretamente a lacuna 10 do texto.

(IME - 2017/2018 - 1 FASE ) Seja o seguinte sistema de equaes, em que S um nmero real: Escolha uma faixa de valores de s em que as solues do sistema so todas negativas.

(IME - 2017/2018 - 1 FASE ) Determine o valor de na expresso abaixo, sabendo-se que 0 a 1, onde Z um nmero complexo que satisfaz a equao: 24.033Z2- 22017Z+1 = 0 Obs.: Im(Z) a parte imaginria do nmero complexo Z

(IME - 2017/2018 - 1 FASE ) Duas partculas A e B carregadas eletricamente com mesmos valores de cargas positivas, partem da origem em velocidade nula no instante t = 0,e tm suas componentes de acelerao em relao aos eixos X e Y regidas pelas seguintes equaes temporais: Partcula A: Partcula B: O instante tmin, onde, em que a fora de repulso entre as cargas mnima

(IME - 2017/2018 - 2 FASE) HYPATIA OF ALEXANDRIA Hypatia of Alexandria (c. 370 CE March 415 CE) was a female philosopher and mathematician, born in Alexandria, Egypt possibly in 370 CE (although some scholars cite her birth as c. 350 CE). She was the daughter of the mathematician Theon, the last Professor at the University of Alexandria, who tutored her in math, astronomy, and the philosophy of the day which, in modern times, would be considered science. Nothing is known of her mother and there is little information about her life. As the historian Deakin writes, The most detailed accounts we have of Hypatias life are the records of her death. We learn more about her death from the primary sources than we do about any other aspect of her life. She was murdered in 415 CE by a Christian mob who attacked her on the streets of Alexandria. The primary sources, even those Christian writers who were hostile to her and claimed she was a witch, portray her as a woman who was widely known for her generosity, love of learning, and expertise in teaching in the subjects of Neo-Platonism, mathematics, science, and philosophy in general. In a city which was becoming increasingly diverse religiously (and had always been so culturally) Hypatia was a close friend of the pagan prefect Orestes and was blamed by Cyril, the Christian Archbishop of Alexandria, for keeping Orestes from accepting the true faith. She was also seen as a stumbling block to those who would have accepted the truth of Christianity were it not for her charisma, charm, and excellence in making difficult mathematical and philosophical concepts understandable to her students; concepts which contradicted the teachings of the relatively new church. Alexandria was a great seat of learning in the early days of Christianity but, as the faith grew in adherents and power, steadily became divided by fighting among religious factions. It is by no means an exaggeration to state that Alexandria was destroyed as a centre of culture and learning by religious intolerance, and 1Hypatia has come to symbolize this tragedy to the extent that her death has been cited as the end of the classical world. 2By all accounts, Hypatia was an extraordinary woman __________. Theon refused to impose upon his daughter the traditional role assigned to women and raised her as one would have raised a son in the Greek tradition; by teaching her his own trade. The historian Slatkin writes, Greek women of all classes were occupied with the same type of work, mostly centered around the domestic needs of the family. Women cared for young children, nursed the sick, and prepared food. Hypatia, on the other hand, led the life of a respected academic at Alexandrias university; a position to which, as far as the evidence suggests, only males were entitled previously. She never married and remained celibate throughout her life, devoting herself to learning and teaching. The ancient writers are in agreement that she was a woman of enormous intellectual power. Deakin writes: The breadth of her interests is most impressive. Within mathematics, she wrote or lectured on astronomy (including its observational aspects - the astrolabe), geometry (and for its day advanced geometry at that) and algebra (again, for its time, difficult algebra), and made an advance in computational technique - all this as well as engaging in religious philosophy and aspiring to a good writing style. 3Her writings were, as best we can judge, an outgrowth of her teaching in the technical areas of mathematics. In effect, she was continuing a program initiated by her father: a conscious effort to preserve and to elucidate the great mathematical works of the Alexandrian heritage. CE = Common Era, the same as AD, Anno Domini. MARK, Joshua J. Hypatia of Alexandria. In: Ancient History Encyclopedia, 2017. Disponvel em: http://www.ancient.eu/Hypatia_of_Alexandria/. Acesso em: 20/06/2017. Choose the correct option.

(IME - 2017/2018 - 2 FASE) Texto 2 Das vantagens de ser bobo O bobo, por no se ocupar com ambies, tem tempo para ver, ouvir e tocar o mundo. O bobo capaz de ficar sentado quase sem se mexer por duas horas. Se perguntado por que no faz alguma coisa, responde: Estou fazendo. Estou pensando. 1Ser bobo s vezes oferece um mundo de sada porque os espertos s se lembram de sair por meio da esperteza, e o bobo tem originalidade, espontaneamente lhe vem a ideia. O bobo tem oportunidade de ver coisas que os espertos no veem. Os espertos esto sempre to atentos _____i_____ espertezas alheias que se descontraem diante dos bobos, e estes os veem como simples pessoas humanas. O bobo ganha utilidade e sabedoria para viver. O bobo nunca parece ter tido vez. No entanto, muitas vezes,2o bobo um Dostoievski. _____ii_____ desvantagem, obviamente. Uma boba, por exemplo, confiou na palavra de um desconhecido para _____iii_____ compra de um ar refrigerado de segunda mo:3ele disse que o aparelho era novo, praticamente sem uso porque se mudara para a Gvea onde fresco. Vai a boba e compra o aparelho sem v-lo sequer. Resultado: no funciona. Chamado um tcnico, a opinio deste era de que o aparelho estava to estragado que o conserto seria carssimo: mais valia comprar outro. Mas, em contrapartida, a vantagem de ser bobo ter boa-f, no desconfiar, e portanto estar tranquilo, enquanto o esperto no dorme noite com medo de ser ludibriado. O esperto vence com lcera no estmago. O bobo no percebe que venceu. Aviso: no confundir bobos com burros. Desvantagem: pode receber uma punhalada de quem menos espera. uma das tristezas que o bobo no prev. Csar terminou dizendo a clebre frase: At tu, Brutus?. Bobo no reclama. Em compensao, como exclama! Os bobos, com todas as suas palhaadas, devem estar todos no cu. Se Cristo tivesse sido esperto no teria morrido na cruz. O bobo sempre to simptico que h espertos que se fazem passar por bobos. Ser bobo uma criatividade e, como toda criao, difcil. Por isso que os espertos no conseguem passar por bobos.4Os espertos ganham dos outros. Em compensao os bobos ganham a vida.5Bem-aventurados os bobos porque sabem sem que ningum desconfie. Alis no se importam que saibam que eles sabem. H lugares que facilitam mais _____iv_____ pessoas serem bobas (no confundir bobo com burro, com tolo, com ftil). Minas Gerais, por exemplo, facilita ser bobo. Ah, quantos perdem por no nascer em Minas! Bobo Chagall, que pe vaca no espao, voando por cima das casas. quase impossvel evitar o excesso de amor que o bobo provoca. que s6o bobo capaz de excesso de amor. E s o amor faz o bobo. LISPECTOR, Clarice.Das vantagens de ser bobo. Disponvel em: http://www.revistapazes.com/das-vantagens-de-ser-bobo/. Acesso em 10 de maio de 2017. Originalmente publicado no Jornal do Brasil em 12 de setembro de 1970. Sobre as consideraes a respeito de ser esperto vs. ser bobo encontradas no texto, assinale o par de anlises que destoa das consideraes feitas pela autora.

(IME - 2017/2018 - 1 FASE ) A menor raiz real positiva da equaoencontra- se no intervalo:

(IME - 2017/2018 - 2 FASE) Texto 2 Das vantagens de ser bobo O bobo, por no se ocupar com ambies, tem tempo para ver, ouvir e tocar o mundo. O bobo capaz de ficar sentado quase sem se mexer por duas horas. Se perguntado por que no faz alguma coisa, responde: Estou fazendo. Estou pensando. 1Ser bobo s vezes oferece um mundo de sada porque os espertos s se lembram de sair por meio da esperteza, e o bobo tem originalidade, espontaneamente lhe vem a ideia. O bobo tem oportunidade de ver coisas que os espertos no veem. Os espertos esto sempre to atentos _____i_____ espertezas alheias que se descontraem diante dos bobos, e estes os veem como simples pessoas humanas. O bobo ganha utilidade e sabedoria para viver. O bobo nunca parece ter tido vez. No entanto, muitas vezes,2o bobo um Dostoievski. _____ii_____ desvantagem, obviamente. Uma boba, por exemplo, confiou na palavra de um desconhecido para _____iii_____ compra de um ar refrigerado de segunda mo:3ele disse que o aparelho era novo, praticamente sem uso porque se mudara para a Gvea onde fresco. Vai a boba e compra o aparelho sem v-lo sequer. Resultado: no funciona. Chamado um tcnico, a opinio deste era de que o aparelho estava to estragado que o conserto seria carssimo: mais valia comprar outro. Mas, em contrapartida, a vantagem de ser bobo ter boa-f, no desconfiar, e portanto estar tranquilo, enquanto o esperto no dorme noite com medo de ser ludibriado. O esperto vence com lcera no estmago. O bobo no percebe que venceu. Aviso: no confundir bobos com burros. Desvantagem: pode receber uma punhalada de quem menos espera. uma das tristezas que o bobo no prev. Csar terminou dizendo a clebre frase: At tu, Brutus?. Bobo no reclama. Em compensao, como exclama! Os bobos, com todas as suas palhaadas, devem estar todos no cu. Se Cristo tivesse sido esperto no teria morrido na cruz. O bobo sempre to simptico que h espertos que se fazem passar por bobos. Ser bobo uma criatividade e, como toda criao, difcil. Por isso que os espertos no conseguem passar por bobos.4Os espertos ganham dos outros. Em compensao os bobos ganham a vida.5Bem-aventurados os bobos porque sabem sem que ningum desconfie. Alis no se importam que saibam que eles sabem. H lugares que facilitam mais _____iv_____ pessoas serem bobas (no confundir bobo com burro, com tolo, com ftil). Minas Gerais, por exemplo, facilita ser bobo. Ah, quantos perdem por no nascer em Minas! Bobo Chagall, que pe vaca no espao, voando por cima das casas. quase impossvel evitar o excesso de amor que o bobo provoca. que s6o bobo capaz de excesso de amor. E s o amor faz o bobo. LISPECTOR, Clarice.Das vantagens de ser bobo. Disponvel em: http://www.revistapazes.com/das-vantagens-de-ser-bobo/. Acesso em 10 de maio de 2017. Originalmente publicado no Jornal do Brasil em 12 de setembro de 1970. Na frase s o bobo capaz de excesso de amor (referncia 6), a semntica da palavra s, nesse contexto,

(IME - 2017/2018 - 2 FASE) HYPATIA OF ALEXANDRIA Hypatia of Alexandria (c. 370 CE March 415 CE) was a female philosopher and mathematician, born in Alexandria, Egypt possibly in 370 CE (although some scholars cite her birth as c. 350 CE). She was the daughter of the mathematician Theon, the last Professor at the University of Alexandria, who tutored her in math, astronomy, and the philosophy of the day which, in modern times, would be considered science. Nothing is known of her mother and there is little information about her life. As the historian Deakin writes, The most detailed accounts we have of Hypatias life are the records of her death. We learn more about her death from the primary sources than we do about any other aspect of her life. She was murdered in 415 CE by a Christian mob who attacked her on the streets of Alexandria. The primary sources, even those Christian writers who were hostile to her and claimed she was a witch, portray her as a woman who was widely known for her generosity, love of learning, and expertise in teaching in the subjects of Neo-Platonism, mathematics, science, and philosophy in general. In a city which was becoming increasingly diverse religiously (and had always been so culturally) Hypatia was a close friend of the pagan prefect Orestes and was blamed by Cyril, the Christian Archbishop of Alexandria, for keeping Orestes from accepting the true faith. She was also seen as a stumbling block to those who would have accepted the truth of Christianity were it not for her charisma, charm, and excellence in making difficult mathematical and philosophical concepts understandable to her students; concepts which contradicted the teachings of the relatively new church. Alexandria was a great seat of learning in the early days of Christianity but, as the faith grew in adherents and power, steadily became divided by fighting among religious factions. It is by no means an exaggeration to state that Alexandria was destroyed as a centre of culture and learning by religious intolerance, and 1Hypatia has come to symbolize this tragedy to the extent that her death has been cited as the end of the classical world. 2By all accounts, Hypatia was an extraordinary woman __________. Theon refused to impose upon his daughter the traditional role assigned to women and raised her as one would have raised a son in the Greek tradition; by teaching her his own trade. The historian Slatkin writes, Greek women of all classes were occupied with the same type of work, mostly centered around the domestic needs of the family. Women cared for young children, nursed the sick, and prepared food. Hypatia, on the other hand, led the life of a respected academic at Alexandrias university; a position to which, as far as the evidence suggests, only males were entitled previously. She never married and remained celibate throughout her life, devoting herself to learning and teaching. The ancient writers are in agreement that she was a woman of enormous intellectual power. Deakin writes: The breadth of her interests is most impressive. Within mathematics, she wrote or lectured on astronomy (including its observational aspects - the astrolabe), geometry (and for its day advanced geometry at that) and algebra (again, for its time, difficult algebra), and made an advance in computational technique - all this as well as engaging in religious philosophy and aspiring to a good writing style. 3Her writings were, as best we can judge, an outgrowth of her teaching in the technical areas of mathematics. In effect, she was continuing a program initiated by her father: a conscious effort to preserve and to elucidate the great mathematical works of the Alexandrian heritage. CE = Common Era, the same as AD, Anno Domini. MARK, Joshua J. Hypatia of Alexandria. In: Ancient History Encyclopedia, 2017. Disponvel em: http://www.ancient.eu/Hypatia_of_Alexandria/. Acesso em: 20/06/2017. Choose the correct option.

(IME - 2017/2018 - 1 FASE ) O sistema mostrado na figura acima encontra-se em equilbrio esttico, sendo composto por seis cubos idnticos, cada um com massa especfica uniformemente distribuda e de aresta a, apoiados em uma alavanca composta por uma barra rgida de massa desprezvel. O comprimento L da barra para que o sistema esteja em equilbrio :

(IME - 2017/2018 - 1 FASE ) Seja uma elipse com focos no eixo OXe centrada na origem. Seus eixos medem 10e 20/3 Considere uma hiprbole tal que os focos da elipse so os vrtices da hiprbole e os focos da hiprbole so os vrtices da elipse. As parbolas que passam pelas intersees entre a elipse e a hiprbole e que so tangentes ao eixo OYna origem, tm as seguintes equaes:

(IME - 2017/2018 - 2 FASE) HYPATIA OF ALEXANDRIA Hypatia of Alexandria (c. 370 CE March 415 CE) was a female philosopher and mathematician, born in Alexandria, Egypt possibly in 370 CE (although some scholars cite her birth as c. 350 CE). She was the daughter of the mathematician Theon, the last Professor at the University of Alexandria, who tutored her in math, astronomy, and the philosophy of the day which, in modern times, would be considered science. Nothing is known of her mother and there is little information about her life. As the historian Deakin writes, The most detailed accounts we have of Hypatias life are the records of her death. We learn more about her death from the primary sources than we do about any other aspect of her life. She was murdered in 415 CE by a Christian mob who attacked her on the streets of Alexandria. The primary sources, even those Christian writers who were hostile to her and claimed she was a witch, portray her as a woman who was widely known for her generosity, love of learning, and expertise in teaching in the subjects of Neo-Platonism, mathematics, science, and philosophy in general. In a city which was becoming increasingly diverse religiously (and had always been so culturally) Hypatia was a close friend of the pagan prefect Orestes and was blamed by Cyril, the Christian Archbishop of Alexandria, for keeping Orestes from accepting the true faith. She was also seen as a stumbling block to those who would have accepted the truth of Christianity were it not for her charisma, charm, and excellence in making difficult mathematical and philosophical concepts understandable to her students; concepts which contradicted the teachings of the relatively new church. Alexandria was a great seat of learning in the early days of Christianity but, as the faith grew in adherents and power, steadily became divided by fighting among religious factions. It is by no means an exaggeration to state that Alexandria was destroyed as a centre of culture and learning by religious intolerance, and 1Hypatia has come to symbolize this tragedy to the extent that her death has been cited as the end of the classical world. 2By all accounts, Hypatia was an extraordinary woman __________. Theon refused to impose upon his daughter the traditional role assigned to women and raised her as one would have raised a son in the Greek tradition; by teaching her his own trade. The historian Slatkin writes, Greek women of all classes were occupied with the same type of work, mostly centered around the domestic needs of the family. Women cared for young children, nursed the sick, and prepared food. Hypatia, on the other hand, led the life of a respected academic at Alexandrias university; a position to which, as far as the evidence suggests, only males were entitled previously. She never married and remained celibate throughout her life, devoting herself to learning and teaching. The ancient writers are in agreement that she was a woman of enormous intellectual power. Deakin writes: The breadth of her interests is most impressive. Within mathematics, she wrote or lectured on astronomy (including its observational aspects - the astrolabe), geometry (and for its day advanced geometry at that) and algebra (again, for its time, difficult algebra), and made an advance in computational technique - all this as well as engaging in religious philosophy and aspiring to a good writing style. 3Her writings were, as best we can judge, an outgrowth of her teaching in the technical areas of mathematics. In effect, she was continuing a program initiated by her father: a conscious effort to preserve and to elucidate the great mathematical works of the Alexandrian heritage. CE = Common Era, the same as AD, Anno Domini. MARK, Joshua J. Hypatia of Alexandria. In: Ancient History Encyclopedia, 2017. Disponvel em: http://www.ancient.eu/Hypatia_of_Alexandria/. Acesso em: 20/06/2017. Choose the correct option.